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Properties

Label 22.0.31867614805...1747.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,947^{21}$
Root discriminant $693.52$
Ramified prime $947$
Class number $18795485$ (GRH)
Class group $[18795485]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10971649548548503, -7067594870875052, 2756828139456442, -92001319669394, 285911612185382, -29371703518119, 11302889577644, -352417867544, 1442099273183, 321183439624, 126935713555, 23061834780, 4884348545, 585651628, 72912964, 3003492, 29342, -68736, -4834, -638, 22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 22*x^20 - 638*x^19 - 4834*x^18 - 68736*x^17 + 29342*x^16 + 3003492*x^15 + 72912964*x^14 + 585651628*x^13 + 4884348545*x^12 + 23061834780*x^11 + 126935713555*x^10 + 321183439624*x^9 + 1442099273183*x^8 - 352417867544*x^7 + 11302889577644*x^6 - 29371703518119*x^5 + 285911612185382*x^4 - 92001319669394*x^3 + 2756828139456442*x^2 - 7067594870875052*x + 10971649548548503)
 
gp: K = bnfinit(x^22 - x^21 + 22*x^20 - 638*x^19 - 4834*x^18 - 68736*x^17 + 29342*x^16 + 3003492*x^15 + 72912964*x^14 + 585651628*x^13 + 4884348545*x^12 + 23061834780*x^11 + 126935713555*x^10 + 321183439624*x^9 + 1442099273183*x^8 - 352417867544*x^7 + 11302889577644*x^6 - 29371703518119*x^5 + 285911612185382*x^4 - 92001319669394*x^3 + 2756828139456442*x^2 - 7067594870875052*x + 10971649548548503, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 22 x^{20} - 638 x^{19} - 4834 x^{18} - 68736 x^{17} + 29342 x^{16} + 3003492 x^{15} + 72912964 x^{14} + 585651628 x^{13} + 4884348545 x^{12} + 23061834780 x^{11} + 126935713555 x^{10} + 321183439624 x^{9} + 1442099273183 x^{8} - 352417867544 x^{7} + 11302889577644 x^{6} - 29371703518119 x^{5} + 285911612185382 x^{4} - 92001319669394 x^{3} + 2756828139456442 x^{2} - 7067594870875052 x + 10971649548548503 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-318676148059321885950263776987138488850287420742260109802771747=-\,947^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $693.52$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $947$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(947\)
Dirichlet character group:    $\lbrace$$\chi_{947}(1,·)$, $\chi_{947}(514,·)$, $\chi_{947}(643,·)$, $\chi_{947}(580,·)$, $\chi_{947}(133,·)$, $\chi_{947}(390,·)$, $\chi_{947}(769,·)$, $\chi_{947}(17,·)$, $\chi_{947}(658,·)$, $\chi_{947}(215,·)$, $\chi_{947}(732,·)$, $\chi_{947}(289,·)$, $\chi_{947}(930,·)$, $\chi_{947}(178,·)$, $\chi_{947}(557,·)$, $\chi_{947}(814,·)$, $\chi_{947}(367,·)$, $\chi_{947}(304,·)$, $\chi_{947}(433,·)$, $\chi_{947}(946,·)$, $\chi_{947}(185,·)$, $\chi_{947}(762,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{15} - \frac{3}{7} a^{14} + \frac{3}{7} a^{13} + \frac{3}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{15} - \frac{2}{7} a^{14} + \frac{1}{7} a^{13} + \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{15} + \frac{3}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7}$, $\frac{1}{287} a^{19} + \frac{3}{287} a^{18} - \frac{3}{287} a^{17} + \frac{11}{287} a^{16} + \frac{97}{287} a^{15} + \frac{64}{287} a^{14} + \frac{20}{287} a^{13} - \frac{60}{287} a^{12} - \frac{117}{287} a^{11} + \frac{15}{287} a^{10} - \frac{14}{41} a^{9} + \frac{20}{41} a^{7} + \frac{92}{287} a^{6} + \frac{45}{287} a^{5} + \frac{143}{287} a^{4} - \frac{135}{287} a^{3} + \frac{43}{287} a^{2} - \frac{97}{287} a + \frac{3}{7}$, $\frac{1}{4617249973} a^{20} - \frac{6425760}{4617249973} a^{19} + \frac{43434791}{4617249973} a^{18} - \frac{254248747}{4617249973} a^{17} - \frac{64952890}{4617249973} a^{16} - \frac{2051817667}{4617249973} a^{15} - \frac{511856621}{4617249973} a^{14} - \frac{10337854}{112615853} a^{13} + \frac{102380917}{659607139} a^{12} + \frac{2244385953}{4617249973} a^{11} - \frac{1085934827}{4617249973} a^{10} - \frac{1472491432}{4617249973} a^{9} - \frac{270232418}{4617249973} a^{8} + \frac{186491759}{4617249973} a^{7} + \frac{729278432}{4617249973} a^{6} - \frac{1126261317}{4617249973} a^{5} - \frac{324929181}{659607139} a^{4} - \frac{40972816}{659607139} a^{3} + \frac{1066492893}{4617249973} a^{2} + \frac{2196138387}{4617249973} a - \frac{3629001}{16087979}$, $\frac{1}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{21} - \frac{9871609577302540885921911180286700418554236218158162260440015984941113823247572148097381203434498116844365209035679753153936194593158747522}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{20} + \frac{158309863281632864678376138207669507728065036068501197919999174698024651592960486209921672559384034205145459538985552567746225684404261970165522234}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{19} - \frac{2371344341152885106848580579231758821056710825052023679549197616718346374139883730700497356486513615692340991729853865027348025051880309544770838924}{42208248220729840564274896464968952660519238531427542699372495003507544118553709333131130863425836544343311613085341707253596298594136600619237454237} a^{18} - \frac{10184954750868592044151124691499231380325240635853453786698737307483672516179693561227223137987264229489514365894690866946630178782596254572478841709}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{17} - \frac{2772302969192720726811742254510789386434542982201581398059742055790877082798974384648775669164834795783346464028084284244490401668183068528577471318}{42208248220729840564274896464968952660519238531427542699372495003507544118553709333131130863425836544343311613085341707253596298594136600619237454237} a^{16} - \frac{108950201607353609561398505323274415428701021906310850235133681187049471776195928935168604009146136621432593992267799290750466127410630538388552420824}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{15} + \frac{126918002775619958095061906558518757129144843399407592656849752453071306015867770396450525476518953449899069487984305882911525372843195406866527405279}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{14} + \frac{92146345797678012653855145047660446000496300484841927771843303340371582005578655555930979903834760459959931616113601562788499260813071969772599506931}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{13} + \frac{11958474035154058946670059218479553469158591604583612996217739256819672477511303228742386759497459924593970762224143620423998035427095031866511255548}{42208248220729840564274896464968952660519238531427542699372495003507544118553709333131130863425836544343311613085341707253596298594136600619237454237} a^{12} - \frac{123925850383970846797906149653393849796207844445220495482347777108145259767590707543533084650512277422966952961308646315489875084020748047018957949183}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{11} - \frac{100633473160056054022236013624155406695953486059339696989405061760427066495267085057407142075049919862446533540797091308873198250372193363290118474979}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{10} - \frac{75331303664660247385526436416443573952293934320927209920982077442555089533351464841174679162301935265439584104879344291920726218898395477997653325638}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{9} + \frac{88129556367211910434028959082704047078214832901147970548905654278484461682656859326840091024778228688240157704400090953475605746421143866724904815480}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{8} + \frac{754807655256617191014124351557154889135116070804583049355833242572816343650446743398086609359422310427023275723202072965110485811139037956568968196}{2255402576680220488167360879807501287203318089465593884699293626141624494884549353678762717892983632140482299935857953822711252596633253468203528089} a^{7} - \frac{9300678110171402110972542564977255984932585650845521328478991586301649355490598712223855248198239257171809807910783866993753196368266089648004205885}{42208248220729840564274896464968952660519238531427542699372495003507544118553709333131130863425836544343311613085341707253596298594136600619237454237} a^{6} - \frac{42804515879363096984503550602593117211219906870770909445606512201925313999602002581622957617504520273573840203667738865101093280023249637111252599676}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{5} - \frac{9762616949526760835045880097817002110491866657405754166621116257765862165700023476039347170832176616507338083284462063166363616768224910752458501549}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{4} + \frac{75002779718731100124533116993042103255729135771579024768278693757589709185097339115359370762588156500367399880035915294033806704208304989918305330599}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{3} + \frac{93618412963818806243196574661767724999951378489622546316355462899906113197511124722886171332397772256654253537747953529636246552268955617904956346554}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a^{2} + \frac{92264795135257184405278683816318469232899048633285216027628450222813033135561335351199671039205572940788388158094635076289948107453686952778902539106}{295457737545108883949924275254782668623634669719992798895607465024552808829875965331917916043980855810403181291597391950775174090158956204334662179659} a - \frac{3379575554487308482895525430899517005628480977348251223380253742015134301150792165477038242474757618559687161318184425195888825499653362879927682021}{7206286281588021559754250615970308990820357798048604851112377195720800215362828422729705269365386727083004421746277852457931075369730639130113711699}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{18795485}$, which has order $18795485$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1130622785561068.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-947}) \), 11.11.580095892065127629623589183049.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{11}$ $22$ $22$ $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ $22$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
947Data not computed